US8553789B2 - Method for the estimation of OFDM parameters by adaptation of covariance - Google Patents
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L27/00—Modulated-carrier systems
- H04L27/26—Systems using multi-frequency codes
- H04L27/2601—Multicarrier modulation systems
- H04L27/2647—Arrangements specific to the receiver only
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L27/00—Modulated-carrier systems
- H04L27/0012—Modulated-carrier systems arrangements for identifying the type of modulation
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L27/00—Modulated-carrier systems
- H04L27/26—Systems using multi-frequency codes
- H04L27/2601—Multicarrier modulation systems
- H04L27/2602—Signal structure
- H04L27/2605—Symbol extensions, e.g. Zero Tail, Unique Word [UW]
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L27/00—Modulated-carrier systems
- H04L27/26—Systems using multi-frequency codes
- H04L27/2601—Multicarrier modulation systems
- H04L27/2647—Arrangements specific to the receiver only
- H04L27/2655—Synchronisation arrangements
- H04L27/2666—Acquisition of further OFDM parameters, e.g. bandwidth, subcarrier spacing, or guard interval length
Definitions
- This invention relates to a method for estimating OFDM (Orthogonal Frequency Division Multiplex) modulation parameters. It can be applied particularly in opportunistic telecommunication systems (Cognitive Radio).
- OFDM modulation is well known in prior art and is used in many telecommunication systems such as DVB-T, ADSL, Wi-Fi (IEEE 802 a/g), WiMax (IEEE 802.16). It has the advantage of good spectral efficiency and good protection against frequency-selective fading.
- the information symbols to be transmitted are grouped by blocks of N symbols, where N is generally a power of 2, the OFDM symbols being obtained by carrying out an IFFT (Inverse Fast Fourier Transform) on said blocks of information symbols.
- IFFT Inverse Fast Fourier Transform
- a cyclic prefix is added at the beginning of each OFDM symbol in order to absorb the intersymbol interference or ISI and facilitate equalization at reception.
- the OFDM signal comprised of these OFDM symbols can be then frequency up-converted.
- the signal transmitted by an OFDM system can be represented in base band by:
- E is the power of the signal
- N is the number of carriers of the OFDM multiplex
- a k,n are the information symbols relative to the block k, belonging to a M-order modulation alphabet, typically BPSK, QPSK or QAM
- 1/T c is the flow of the information symbols where T c is the “chip” time
- N+D total duration
- T prefix duration guard interval
- the cyclic prefix is a copy of the end of the OFDM symbol into the guard interval.
- the cyclic prefixes are simply omitted, in other words the useful portions of the symbols are separated by “empty” guard intervals. This transmission technique also makes it possible to remove the intersymbol interference but makes the equalization of the signal more complex.
- h s a is the convolution between the OFDM signal transmitted
- s a (t) is the impulse response of the transmission channel h(t)
- b(t) is a random function describing the noise.
- FIG. 2 diagrammatically shows the structure of an OFDM receiver.
- the signal received is sampled in 210 at the chip frequency, then the samples are subjected to a serial/parallel conversion in 220 in order to form blocks of N+D samples.
- the first D samples corresponding to the guard interval are rejected and the block of the N remaining samples corresponding to the useful portion of the OFDM symbol is subjected to a FFT in 230 .
- the demodulated symbols obtained are then subjected to a serial conversion in 240 .
- this receiver requires a precise synchronisation in time and in frequency. Indeed, it is understood that a poor synchronisation in time will result in a progressive time sliding of the truncation window and a poor synchronisation in frequency, a phase rotation of the samples, which can be represented by a multiplication factor e 2i ⁇ fnT c where ⁇ f if the frequency offset between the demodulation frequency of the receiver and the carrier frequency of the OFDM multiplex.
- the time and frequency synchronization of the receiver is generally carried out thanks to the acquisition of a training sequence.
- Modulation parameters or more simply OFDM parameters, here means the useful duration NT c , the duration of the guard interval DT c and/or the repetition period (N+D)T c of these symbols, or the inverse of these values. Note that with regards to this, 1/NT c shows the spacing between sub-carriers and 1/(N+D)T c the symbol frequency.
- the receiver does not a priori know the OFDM modulation parameters and must therefore estimate them blindly before being able to demodulate the signal.
- opportunistic radio systems are known of which a description can be found in the founding thesis of J. Mitola entitled “Cognitive radio: an integrated agent architecture for software defined radio ⁇ , Royal Institute of Technology, Swiss, PhD Dissertation, 8 May 2000.
- a primary user of such a system uses an OFDM modulation (WiFi, Wi-Max, LTE, WRAN, 802.22, DVB-T etc.)
- a secondary user still referred to as opportunistic, must be in a position to detect whether or not an OFDM signal is present in a given band, and consequently to estimate an OFDM parameter of such a signal.
- CSI Channel State Information
- the purpose of this invention is consequently to propose a method for estimating modulation parameters of an OFDM signal, with a high success rate, even with a low signal-to-noise ratio and this, whether or not the signal is devoid of a prefix.
- a subsidiary purpose of this invention is to allow a receiver to carry out a joint estimation of the OFDM parameters and of the signal-to-noise ratio.
- This invention is defined by a method of estimating at least one modulation parameter of an OFDM signal, said signal being sampled during a time window in order to provide a sequence of samples, wherein:
- the theoretical covariance matrix is obtained according to said tested value of the modulation parameter, of a tested value of the power of the signal and of a tested value of the noise power, the steps (c) and (d) then being iterated for a plurality of tested values of signal and noise power.
- the method of estimating can also provide an estimation of the signal-to-noise ratio using values of signal and noise power minimising said distance together with the estimated value of the modulation parameter.
- ⁇ tilde over (T) ⁇ c is the useful duration of the OFDM symbols, known or tested
- ⁇ tilde over (D) ⁇ tilde over (T) ⁇ c is the duration of the prefix of the OFDM symbols, known or tested
- T e is the sampling period of the OFDM signal.
- the covariance matrix can be estimated by:
- T is the k th vector of said plurality of vectors
- y k,p y a (pT e + ⁇ tilde over (D) ⁇ tilde over (T) ⁇ c +k( ⁇ tilde over (T) ⁇ c + ⁇ tilde over (D) ⁇ tilde over (T) ⁇ c ))
- ⁇ tilde over (T) ⁇ c is the useful duration of the OFDM symbols, known or tested
- ⁇ tilde over (D) ⁇ tilde over (T) ⁇ c is the duration of the prefix of the OFDM symbols, known or tested
- T e is the sampling period of the OFDM signal and
- the distance between the covariance matrix and the theoretical covariance matrix can be calculated by means of:
- J MCOMET ⁇ ( ⁇ ⁇ , ⁇ ⁇ a , ⁇ ⁇ b ) K ⁇ P ⁇ 2 ⁇ ⁇ R ⁇ yy ⁇ ( ⁇ ⁇ ) - R yy ⁇ ( ⁇ ⁇ , ⁇ ⁇ a , ⁇ ⁇ b ) ⁇ F 2 where ⁇ . ⁇ F is the Frobenius norm.
- the modulation parameter can be the total duration of an OFDM symbol, the useful duration of an OFDM symbol, the duration of the prefix of an OFDM symbol, the number of sub-carriers of an OFDM symbol.
- Said method of estimating can also be applied to a set of modulation parameters comprised of the useful duration of an OFDM symbol, of the prefix duration of an OFDM symbol and of the number of sub-carriers of the OFDM multiplex, the estimated values of these parameters then being the tested values jointly minimising said distance.
- FIG. 1 diagrammatically shows an OFDM signal
- FIG. 2 diagrammatically shows an OFDM receiver of prior art
- FIG. 3 shows a flow chart of the method of estimating an OFDM modulation parameter according to the invention
- FIG. 4 shows a flow chart of the method of estimating OFDM modulation parameters according to a first embodiment of the invention
- FIG. 5 shows a first example of a cost function used in the method of FIG. 4 ;
- FIG. 6 shows a second example of a cost function used in the method of FIG. 4 ;
- FIG. 7 shows a first example of a standardised cost function used in a second embodiment of the invention.
- FIG. 8 shows a second example of a standardised cost function used in the second embodiment of the invention.
- FIG. 9 gives the good estimate rate of an OFDM modulation parameter according to the signal-to-noise ratio, in the case of a joint estimation
- FIG. 10 gives the good estimate rate of the variance of the signal according to this same ratio, in the case of a joint estimation
- FIG. 11 compares several methods for estimating an OFDM parameter according to the signal-to-noise ratio, in the case of a perfect synchronisation of the receiver and for a short prefix duration;
- FIG. 12 compares several methods for estimating an OFDM parameter according to the signal-to-noise ratio, in the case of a perfect synchronisation of the receiver and in the absence of a prefix;
- FIG. 13 compares several methods for estimating an OFDM parameter according to the signal-to-noise ratio, in the absence of synchronisation of the receiver and in the absence of a prefix.
- the signal received is sampled at a frequency
- T f e 1 T e , higher than the width of the OFDM band under consideration in order to satisfy the Nyquist criterion. As such, it is certain that the sampling period T e is less than the chip period T c .
- T 0 be the duration of the observation window, the discrete-time signal obtained is composed of
- M 0 ⁇ T 0 T e ⁇ samples (where ⁇ x ⁇ designates the largest integer less than x).
- y k Ga k +b (4)
- a k is the k-th vector of size N of the symbols transmitted
- b is a vector of size P, representing the noise, supposedly additive and Gaussian
- y k is a vector of size P comprised of P consecutive samples y k
- p y a (pT e +DT c +k(NT c +DT c ))
- p 0, . . . , P ⁇ 1.
- the P samples of noise are random variables, independent amongst themselves, and of the useful signal
- G HF where H is the matrix of size N ⁇ N representative of the frequency response of the canal and F is a matrix of size P ⁇ N of which the elements are given by:
- y ⁇ tilde over ( ⁇ ) ⁇ k is a vector of ⁇ tilde over (P) ⁇ random variables of which the covariance matrix equals:
- ⁇ a 2 and ⁇ b 2 show respectively the variance of the symbols transmitted and of the noise
- I ⁇ tilde over ( ⁇ ) ⁇ is the unit matrix of size ⁇ .
- the ratio ⁇ a 2 / ⁇ b 2 is no other than the signal-to-noise ratio.
- the covariance matrix depends in general on the parameter ⁇ tilde over ( ⁇ ) ⁇ and on the respective powers of the useful signal and of the noise.
- the idea at the base of this invention is to estimate at least one modulation parameter of the OFDM signal by using a method for adapting the covariance.
- FIG. 3 generally shows a flow chart of the method of estimating an OFDM modulation parameter according to the invention.
- the OFDM signal is sampled in base-band during a time window of a given length. A sequence of samples is thus obtained.
- samples of the sequence thus obtained are grouped into packets, according to a tested value, ⁇ tilde over ( ⁇ ) ⁇ , of the parameter to be estimated, each packet being formed by samples consecutive of said sequence and being represented by a vector noted y ⁇ tilde over ( ⁇ ) ⁇ k .
- the covariance matrix ⁇ circumflex over (R) ⁇ yy ( ⁇ tilde over ( ⁇ ) ⁇ ) of the vectors thus obtained is estimated.
- the theoretical covariance matrix R yy ( ⁇ tilde over ( ⁇ ) ⁇ ) is then calculated which would be obtained using an OFDM signal of which said modulation parameter would have said tested value ⁇ tilde over ( ⁇ ) ⁇ .
- a distance is calculated between the covariance matrix in 330 and the theoretical covariance matrix obtained in 340 .
- the steps 320 to 350 are repeated for a plurality of possible values of the parameter ⁇ tilde over ( ⁇ ) ⁇ .
- the estimated value, ⁇ tilde over ( ⁇ ) ⁇ , of this parameter is obtained in 360 as that minimising the distance calculated in the step 350 .
- FIG. 4 shows in a more detailed manner the method of estimating at least one OFDM modulation parameter according to a first embodiment of the invention.
- the signal received is sampled in base-band, y, during a time window of width T 0 .
- the sampling period T e is selected sufficiently short enough in order to comply with the Nyquist criterion, in light of the length of the OFDM band at hand. A sequence of M 0 samples is thus obtained.
- a value to be tested of the parameter ⁇ tilde over ( ⁇ ) ⁇ is chosen, i.e. values of useful duration ⁇ tilde over (T) ⁇ c , of prefix duration ⁇ tilde over (T) ⁇ c , and of the number of sub-carriers ⁇ .
- the ⁇ tilde over (P) ⁇ components y k,p , p 0, . . .
- the tested value of the modulation parameter, ⁇ tilde over ( ⁇ ) ⁇ determines the size of these vectors as well as their starting points in the sequence.
- the covariance matrix R yy ( ⁇ tilde over ( ⁇ ) ⁇ ) is estimated by calculating:
- J COMET min J COMET ( ⁇ tilde over ( ⁇ ) ⁇ , ⁇ tilde over ( ⁇ ) ⁇ a , ⁇ tilde over ( ⁇ ) ⁇ b )>J COMET min
- J COMET min J COMET ( ⁇ tilde over ( ⁇ ) ⁇ , ⁇ tilde over ( ⁇ ) ⁇ a , ⁇ tilde over ( ⁇ ) ⁇ b ) and corresponding values ( ⁇ tilde over ( ⁇ ) ⁇ , ⁇ tilde over ( ⁇ ) ⁇ a , ⁇ tilde over ( ⁇ ) ⁇ b ) are stored in a memory.
- This ordered set is for example the Cartesian product of basic sets ⁇ ⁇ ⁇ ⁇ a ⁇ ⁇ b where ⁇ ⁇ is a set of possible values of the parameters provided by an OFDM standard, and ⁇ ⁇ a , ⁇ ⁇ b are intervals of values of power.
- the order relation on this set can be the lexicographic order.
- the values of the parameters ⁇ , ⁇ a , ⁇ b are estimated using:
- the method of estimating shown in FIG. 4 makes it possible to jointly estimate the modulation parameters of the OFDM signal (represented by ⁇ ) as well as the signal and noise power levels.
- This method consequently makes it possible not only to determine NT c , DT c and N but also for example the signal-to-noise ratio ⁇ a 2 / ⁇ b 2 .
- ⁇ tilde over (D) ⁇ tilde over (T) ⁇ c DT c
- ⁇ N
- ⁇ tilde over ( ⁇ ) ⁇ a ⁇ a
- ⁇ tilde over ( ⁇ ) ⁇ b ⁇ b .
- the signal-to-noise ratio ⁇ a 2 / ⁇ a b here is equal to 10 dB.
- the cost function would gain have a pronounced minimum for the corresponding actual value.
- J MCOMET ⁇ ( ⁇ ⁇ , ⁇ ⁇ a , ⁇ ⁇ b ) J COMET ⁇ ( ⁇ ⁇ , ⁇ ⁇ a , ⁇ ⁇ b ) E ⁇ ( J COMET ⁇ ( ⁇ ⁇ , ⁇ ⁇ a , ⁇ ⁇ b ) ) ( 15 )
- E(J COMET ( ⁇ tilde over ( ⁇ ) ⁇ , ⁇ tilde over ( ⁇ ) ⁇ a , ⁇ tilde over ( ⁇ ) ⁇ b )) designates the average of the cost function. It can be shown that for the low values of the signal-to-noise ratio, i.e. for ⁇ a ⁇ b , this average is equal to:
- J MCOMET ⁇ ( ⁇ ⁇ , ⁇ ⁇ a , ⁇ ⁇ b ) K ⁇ P ⁇ 2 ⁇ J COMET ⁇ ( ⁇ ⁇ , ⁇ ⁇ a , ⁇ ⁇ b ) ( 17 )
- the method for estimating according to the second embodiment of the invention differs from that shown in FIG. 4 in that the cost function J MCOMET is used in place of J COMET .
- the description of the processing steps therefore shall not be included here.
- FIG. 7 shows a first example of a normalised cost function, in the same conditions as those in FIG. 5 .
- FIG. 8 shows a first example of a normalised cost function, in the same conditions as those in FIG. 6 .
- the normalised cost function is convex and has a pronounced minimum in both cases, in particular at a low level of signal-to-noise ratio, cf. FIG. 8 .
- the method of estimating according to the second embodiment is consequently robust with respect to noise.
- FIG. 9 shows the rate of good estimate (or of identification) of the parameter NT c according to the signal-to-noise ratio, in the case of a joint estimation of NT c , ⁇ a and ⁇ b .
- the rate of good estimate is equal to 1 with the normalised cost function J MCOMET when the signal-to-noise ratio is greater than ⁇ 8 dB.
- this rate of good estimate is achieved with the cost function J COMET only for a signal-to-noise ratio greater than 2 dB.
- FIG. 10 shows the rate of good estimate of the variance ⁇ a according to the signal-to-noise ratio, in the case of a joint estimation of NT c , ⁇ a and ⁇ b . Note here again that a rate of good estimate equal to 1 is achieved starting at ⁇ 8 dB for J MCOMET and only starting at 2 dB for the cost function J COMET .
- FIGS. 11 , 12 and 13 compare the performance of different methods for estimating an OFDM parameter, here the parameter NT c , in terms of the rate of good estimate, according to the signal-to-noise ratio.
- I and II designate the curves relative to the known methods for estimation, based respectively on the correlation and the cyclic correlation of the signal received, such as described in the article of A. Bouzezgi et al. entitled “A second order statistics based algorithm for blind recognition of OFDM based systems” published in IEEE Global Telecommunications Conference, November 2008, and III and IV designate the curves relative respectively to the second and to the first embodiments of the invention.
- the method of estimating according to the invention can be implemented in an OFDM receiver or a receiver of a secondary user in an opportunistic radio system, using dedicated circuits or software modules executed by a microprocessor, in a manner known per se.
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Description
where E is the power of the signal, N is the number of carriers of the OFDM multiplex, ak,n are the information symbols relative to the block k, belonging to a M-order modulation alphabet, typically BPSK, QPSK or QAM, 1/Tc is the flow of the information symbols where Tc is the “chip” time, Ts is the total duration of the OFDM symbol with Ts=(N+D)Tc, where D is the size of the cyclic prefix expressed as a number of chips, ga(t) is the impulse response of the shaping filter of the OFDM symbols, of time support [0,Ts], intended to apodise the spectrum of the signal.
y a(t)=h s a(t)+b(t) (2)
where hsa is the convolution between the OFDM signal transmitted, sa(t) is the impulse response of the transmission channel h(t), and b(t) is a random function describing the noise. We shall suppose that the length of the impulse response of the canal is less than the duration of the guard interval, in such a way that the intersymbol interference (ISI) can be left aside.
z k,n =h n a k,n +b k,n (3)
where hn is a complex coefficient which depends on the impulse response of the transmission channel, and bk,n is a random variable representing a sample of noise.
R yy({tilde over (θ)},{tilde over (σ)}a,{tilde over (σ)}b)={tilde over (σ)}a 2 F {tilde over (θ)} F {tilde over (θ)} H+{tilde over (σ)}b 2 I {tilde over (θ)}
where {tilde over (θ)}, {tilde over (σ)}a 2, {tilde over (σ)}b 2 are respectively the tested values of the modulation parameter, of the signal power and of the noise power; I{tilde over (θ)} is the unit matrix of size Ñ×Ñ where Ñ is the number of sub-carriers of the OFDM multiplex, known or supposed, and the matrix F{tilde over (θ)} is a matrix of size {tilde over (P)}×Ñ with {tilde over (P)}=└(Ñ{tilde over (T)}c+{tilde over (D)}{tilde over (T)}c)/{tilde over (T)}e┘, of which the elements are given by:
where Ñ{tilde over (T)}c is the useful duration of the OFDM symbols, known or tested, {tilde over (D)}{tilde over (T)}c is the duration of the prefix of the OFDM symbols, known or tested, and Te is the sampling period of the OFDM signal.
where yk=(yk,0, . . . yk,{tilde over (P)}-1)T is the k th vector of said plurality of vectors, yk,p=ya(pTe+{tilde over (D)}{tilde over (T)}c+k(Ñ{tilde over (T)}c+{tilde over (D)}{tilde over (T)}c)) where Ñ{tilde over (T)}c is the useful duration of the OFDM symbols, known or tested, {tilde over (D)}{tilde over (T)}c is the duration of the prefix of the OFDM symbols, known or tested, Te is the sampling period of the OFDM signal and
with {tilde over (P)}=└(Ñ{tilde over (T)}c+{tilde over (D)}{tilde over (T)}c)/{tilde over (T)}e┘ and M is the total number of samples in the time window.
J COMET({tilde over (θ)},{tilde over (σ)}a,{tilde over (σ)}b)=∥{circumflex over (R)} yy({tilde over (θ)})−R yy({tilde over (θ)},{tilde over (σ)}a,{tilde over (σ)}b)∥F 2
where ∥.∥F is the Frobenius norm.
where ∥.∥F is the Frobenius norm.
higher than the width of the OFDM band under consideration in order to satisfy the Nyquist criterion. As such, it is certain that the sampling period Te is less than the chip period Tc. Let T0 be the duration of the observation window, the discrete-time signal obtained is composed of
samples (where └x┘ designates the largest integer less than x).
y k =Ga k +b (4)
where ak is the k-th vector of size N of the symbols transmitted, b is a vector of size P, representing the noise, supposedly additive and Gaussian and yk is a vector of size P comprised of P consecutive samples yk,p=ya(pTe+DTc+k(NTc+DTc)), p=0, . . . , P−1. The P samples of noise are random variables, independent amongst themselves, and of the useful signal; G=HF where H is the matrix of size N×N representative of the frequency response of the canal and F is a matrix of size P×N of which the elements are given by:
With a concern for simplification, but without loss of generality, we shall suppose that in what follows the transmission channel is without fading, in other words the matrix H is equal to the identity matrix to within about a multiplicative coefficient. The relation (4) can then be written in the form:
y k =Fa k +b (6)
We shall now consider the case where the receiver operates blindly. The parameters NTc, DTc and N, characterising the OFDM signal are therefore unknowns that can be grouped together in the form of a parameter to be estimated {tilde over (θ)}=(Ñ{tilde over (T)}c, {tilde over (D)}{tilde over (T)}c, Ñ). The samples of the OFDM signal received by the receiver are grouped together in the form of vectors of size {tilde over (P)}=└(Ñ{tilde over (T)}c+{tilde over (D)}{tilde over (T)}c)/{tilde over (T)}e┘, noted y{tilde over (θ)}k, with:
y {tilde over (θ)}k =F {tilde over (θ)} a {tilde over (θ)}k +b {tilde over (θ)}k (7)
where, in a manner similar to relation (6), a{tilde over (θ)}k is the vector of size Ñ of the symbols transmitted, b{tilde over (θ)}k is a vector of size {tilde over (P)} representing the samples of additive Gaussian noise, and F{tilde over (θ)} the matrix of size {tilde over (P)}×Ñ of which the elements are given by:
R yy({tilde over (θ)},σn,σb)=E(y {tilde over (θ)} y {tilde over (θ)} H)=σa 2 F {tilde over (θ)} F {tilde over (θ)} H+{tilde over (σ)}b 2 I {tilde over (θ)} (9)
where σa 2 and σb 2 show respectively the variance of the symbols transmitted and of the noise, and I{tilde over (θ)} is the unit matrix of size Ñ×Ñ. With these conventions, the ratio σa 2/σb 2 is no other than the signal-to-noise ratio. The covariance matrix depends in general on the parameter {tilde over (θ)} and on the respective powers of the useful signal and of the noise.
of size {tilde over (P)}=└(Ñ{tilde over (T)}c+{tilde over (D)}{tilde over (T)}c)/{tilde over (T)}e┘. The {tilde over (P)} components yk,p, p=0, . . . , {tilde over (P)}−1 of the vector yk are therefore obtained by:
y k,p =y a(pT e +{tilde over (D)}{tilde over (T)} c +k(Ñ{tilde over (T)} c +{tilde over (D)}{tilde over (T)} c)) (10)
with yk=(yk,0, . . . yk,{tilde over (P)}-1)T.
R yy({tilde over (θ)},{tilde over (σ)}a,{tilde over (σ)}b)={tilde over (σ)}a 2 F {tilde over (θ)} F {tilde over (θ)} H+{tilde over (σ)}b 2 I {tilde over (θ)} (12)
where {tilde over (σ)}a 2 is a tested value of the power of the signal and {tilde over (σ)}b 2 is a tested value of the power of noise.
J COMET({tilde over (θ)},{tilde over (σ)}a,{tilde over (σ)}b)=∥{circumflex over (R)} yy({tilde over (θ)})−R yy({tilde over (θ)},{tilde over (σ)}a,{tilde over (σ)}b)∥F 2 (13)
in other words, the values of these parameters are retrieved in the aforementioned memory zone.
in the case where only the parameter Ñ{tilde over (T)}c is supposedly unknown, with the other parameters taking the known values, {tilde over (D)}{tilde over (T)}c=DTc, Ñ=N, {tilde over (σ)}a=σa; {tilde over (σ)}b=σb. The signal-to-noise ratio σa 2/σa b here is equal to 10 dB. Note that the cost function JCOMET is convex and has a pronounced minimum in Ñ{tilde over (T)}c=NTc, which allows for a precise estimation of the useful length of the OFDM symbol. Similarly, if the useful length were known and if we were looking for example for the prefix length or the number of sub-carriers, the cost function would gain have a pronounced minimum for the corresponding actual value.
where E(JCOMET({tilde over (θ)},{tilde over (σ)}a,{tilde over (σ)}b)) designates the average of the cost function. It can be shown that for the low values of the signal-to-noise ratio, i.e. for σa<<σb, this average is equal to:
As σb is a constant, the normalised cost function can be reduced to:
Claims (12)
R yy({tilde over (θ)},{tilde over (σ)}a,{tilde over (σ)}b)={tilde over (σ)}a 2 F {tilde over (θ)} F {tilde over (θ)} H+{tilde over (σ)}b 2 I {tilde over (θ)}
J COMET({tilde over (θ)},{tilde over (σ)}a,{tilde over (σ)}b)=∥{circumflex over (R)} yy({tilde over (θ)})−R yy({tilde over (θ)},{tilde over (σ)}a,{tilde over (σ)}b)∥F 2
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US9564966B1 (en) * | 2015-09-30 | 2017-02-07 | Osram Sylvania Inc. | Reconstructing light-based communication signals using an alias frequency |
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US20110228830A1 (en) | 2011-09-22 |
FR2954024B1 (en) | 2017-07-28 |
EP2334021A1 (en) | 2011-06-15 |
EP2334021B1 (en) | 2019-01-09 |
JP2011125023A (en) | 2011-06-23 |
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